See Differentialgleichungen, by E. Kamke, pp. 21-23. There is as yet no general solution for the regular Riccati ODE.

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For h&ApplyFunction;x&equals;0, we have an ODE of Bernoulli type which can be solved in a direct manner by dsolve (see odeadvisor,Bernoulli). In the other cases, a good strategy might be to introduce an appropriate change of variables. A general transformation for canceling h&ApplyFunction;x is not yet known.

Depending on the case, the system might be able to solve the problem using one of the transformations above; the above transformations can also be used as departure point for guessing another transformation suitable for the given problem. It is also possible to introduce a change of variables leading to a linear ODE of the second order (see below).

Concerning Riccati ODEs of Special type, the solving scheme can be summarized as follows. The first thing worth noting is that, for c&equals;0 the system succeeds in solving the ODE:

Now, for c≠0, Riccati Special ODEs can be reduced, step by step, to the case c&equals;0, provided that c can be written as cn&equals;−4&InvisibleTimes;n2&InvisibleTimes;n−1 (integer n). Examples of possible values for c:

The idea is to change variables so as to obtain another ODE of type Riccati Special, but with c[n->0], until reaching c0&equals;0. There are two variable transformations leading to the desired reduction, depending on the sign of n in cn above (see examples at the end).